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Given a set $\mathcal{P}$ of $n$ points in the euclidean plane, how can a set $\mathcal{V}$ of $n$ vectors be determined, that has minimal length-sum and renders $\lbrace p_i+v_i\rbrace$ to be the cen …

I am looking for proofs of or counterexamples to the following assumptions: if the corners of a triangle are chosen from a compact subset $\mathcal{S}$ of the Euclidean plane then all three corners …

No, it is not always possible, because it is not always possible to partition the (cyclic) sequence of edge-length into 3 portions of adjacent edges, for whose sums the triangle inequality holds; e.g. …

Question: what is the probability that four distinct points in general position in the Euclidean plane are in convex configuration, depending on the number of leaf nodes in their Minimum Spa …

Is there an algorithm for reporting all empty ellipses, that are locked by a finite set $\mathcal{P}$ of points in the euclidean plane? An ellipse is considered empty, if no inner point is an elem …

Question: Is there already a name for polylines in the euclidean plane, that have the property, that no interior of none the triangles, defined by one of the polyline's endpoints and a non-a …

let a convex polytope $\mathcal{P}$ in $E^n$ be defined as in the tag-description with the additional requirement that their volume be strictly positive. let further the Voronoi Cells $VC(f)$ of $\mat …

Let $P$ be a closed polygon defined by the sequence $p_0,\,\dots,\,p_{n-1},p_0$ of points. Question: how can one construct, with straightedge and compass alone, another sequence of points $q_0,\,\dot …

Question: what is known about the sequence $\mathbb{X}\subset \mathbb{N}_0$ such that for each $k\in \mathbb{X}$ there exists a set of $n$ points in general position in the Euclidean plane such that t …

A complete symmetric graph with $n=4$ vertices, i.e. a $K_4$ is the disjoint union of three perfect matchings $M_{\text{min}},M_{\text{mid}},M_{\text{max}}$ of which $M_{\text{min}}$ denotes the light …

This should rather be a comment, posting it as an answer is to give a visual clue that there is something in the vein of a proof. Elaborating on Timothy Chow's insightful comment it suffices to consid …

Given a finite set $P$ of $n\gt d+1$ points in $d$-dimensional euclidean space. Under the assumption that the points of $P$ are in general position in the sense that $\lbrace p_{i_1},\dots,p_{i_d}\rbr …

Have these kinds of geometric spanning trees already been described? all three are Minimum Spanning Trees of points that are elements of open disks, the only difference being in how the edge-weight …

I have implemented an algorithm that filters the edges of simple complete graph with weighted edges according to a criterion that is inspired by elementary planar geometry and, to my surprise, in the …

consider the set of lines defined by all pairs of points $\lbrace[(u,v),(u-v,v+u)],\ u,v\in\mathbb{N}\rbrace$ in the euclidean plane Question: what is kown about the density of the set of intersectio …